Analyzing the earth conductivity and permittivity frequency dependence influence to electromagnetic transient phenomena
ABSTRACT In this article the quasimodes model is used to observe the
influence, in electromagnetic transient phenomena, of considering a more
accurate representation of soil, taking into account the earth
conductivity and permittivity frequency dependence. For an actual 440 kV
threephase transmission line the soil behavior is represented through a
unique real value of conductance and through a more accurate model,
considering its electromagnetic behavior

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Conference Paper: Polarity and SLG fault rapid detection using waves propagation phenomena in radial distribution network
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ABSTRACT: Wave propagation laws are applied in MW cable networks in order to determine zone of constant polarity in afterfault transients. An “all current” polarity law permitting rapid fault detection is proposed and analysed using several EMTP cable models.Electricity Distribution, 2005. CIRED 2005. 18th International Conference and Exhibition on; 07/2005  SourceAvailable from: Maria Cristina Dias Tavares
Conference Paper: Influence of accurate soil representation for transmissionline parameters: Analyses based on Carson's modified formulations
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ABSTRACT: In this paper the influence of earth's conductivity and permittivity frequency dependence are evaluated when calculating transversal and longitudinal transmission lines' parameters directly from numerical integration of Carson's modified expressions. As an example of the importance of properly considering the frequencydependent soil model, an actual 440 kV single threephase transmissionline was represented, comparing the longitudinal and transversal parameters considering the earth's conductivity and permittivity frequency dependence soil model in relation to the common soil representation with a constant conductivity and permittivity that can be neglected assuming a low frequency approximation.Transmission and Distribution Conference and Exposition: Latin America (T&DLA), 2010 IEEE/PES; 12/2010
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International Conference on Power Systems Transients – IPST 2003 in New Orleans, USA
Influence of Earth Conductivity and Permittivity Frequency
Dependence in Electromagnetic Transient Phenomena
Carlos Medeiros Portela1, Maria Cristina Tavares2 and José Pissolato Filho2
(1) COPPE – Federal University of Rio de Janeiro, Rua Eng. Cesar Grillo, 249, Rio de Janeiro, RJ,
22640150, BRAZIL email : portelac@ism.com.br), (2) School of Electrical and Computer Engineering,
Dept.of Energy Control and Systems, State University of Campinas, PO Box 6106, 13081970, Campinas,
SP, Brazil (email:cristina@dsce.fee.unicamp.br, pisso@dsce.fee.unicamp.br),
The content of the paper is as follows : in Section 2
the soil electromagnetic behavior is described, with the
presentation of some measured results; in Section 3 line
parameters are calculated; in Section 4 an application to
an actual transmission line is presented.
Abstract – In this article a more accurate representation of
soil behavior is presented. The proposed model takes into
account the earth conductivity frequency dependence and
the earth permittivity, which normally are not considered.
One of the aspects covered in the paper is the importance
of properly considering the earth’s electromagnetic behav
ior when calculating transmission line parameters. For an
actual 440 kV threephase transmission line the soil behav
ior is represented through a unique real value of conduc
tance ( the normal approach ) and through the proposed
model.
II. SOIL ELECTROMAGNETIC BEHAVIOR
One essential aspect of grounding systems study and
simulation is adequate soil modeling.
Except for very high electric fields, that originate sig
nificant soil ionization, soil electromagnetic behavior is
essentially linear, but with electric conductivity, σ , and
electric permittivity, ε , strongly frequency dependent.
The magnetic permeability, µ , is, in general, almost
equal to vacuum magnetic permeability, µ0. For slow
variation of electromagnetic entities, a hysteresis type
behavior may occur. For direct current or very slow
variations of electromagnetic entities, humidity migra
tion phenomena, including electroosmosis and effects of
temperature heterogeneity may take place, which cannot
be dealt with only by means of local soil parameters.
For fast transients, namely those associated to light
ning, the soil behavior is important in a reasonably wide
frequency range, typically from 0 to 2 MHz .
Keywords : Soil model, Line parameters, Frequency depend
ence, Electromagnetic transients.
I. INTRODUCTION
One essential aspect of transmission line modeling is
the adequate representation of ground, which has a big
influence in line parameters, ahead of being a dominant
aspect for analysis and for design of line grounding sys
tem. By historical and cultural reasons, most used pro
cedures assume that the ground may be considered as
having a constant conductivity, frequency independent,
and an electric permittivity that can be neglected
( ω ε << σ ). These two assumptions are quite far from
reality, and can originate inadequate line modeling.
In the present paper a new soil model is presented.
This satisfies the physical coherence conditions concern
ing the relation between conductivity (σ) and permittiv
ity (ε) in the frequency domain. Some examples of
measured ground parameters are presented. The effect of
the soil behavior in some transmission line transients
and its influence on the overvoltages obtained are dis
cussed.
For an actual 440 kV threephase transmission line
the soil behavior is represented through a unique real
value of conductance, the most common assumption,
and through a more accurate model of its electromag
netic behavior in relation to the earth conductivity and
permittivity frequency dependence.
In some cases, a proper earth model can lead to very
different results than the ones obtained with a simple
real conductivity value, as shown. The influence of the
frequency dependence of the soil parameters, in some
line transient phenomena, is analyzed.
A. Field Measurement Procedure
Field measurement procedures have been chosen after
measurement tests covering a large number of soil struc
tures and conditions. The basic aspects related to col
lecting the samples are due to the necessity of [15]:
 Assuring maintenance of natural soil consistence
and humidity, with sample material “identical”
to natural ground.
 Avoiding influence of small depth surface effects,
such as sun, wind and vegetation. These effects
may originate an important dispersion, in time
and space, and special measurement difficulties.
To consider such effects correctly, special meth
ods, considering statistical distribution with
space and time correlation, may be required. In
most applications the error resulting of neglect
ing such effects is relatively small.
 Avoiding important effects of local soil heteroge
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International Conference on Power Systems Transients – IPST 2003 in New Orleans, USA
neity.
 Limit measurement errors related to electrode
shape and contact conditions between electrodes
and soil material.
Three basic procedures where adopted, namely :
 For rock, cylindrical samples (with 0.1 m diame
ter with 0.8 m length) are obtained with a boring
machine.
 For reasonably consistent soils, a cutting and col
lecting procedure is applied, obtaining samples
with a cuboid shape (1.2 m x 0.2 m x 0.2 m)
which are covered with a net, paraffin and a
wood box.
 For sand an pulvurulent soil, samples are col
lected with a plastic tube with diameter 0.2 m
and 1.2 m long, to which steel pieces are
adapted to obtain easy penetration in soil and
sample cutting in tube extremity.
Two current copper plate electrodes (CE) are adapted
at sample extremities (with adjusted pressure) and two
copper cylindrical voltage electrodes (VE) are inserted,
with exemplificative geometry as in Figure 1. Through
an oscillator with variable frequency f , it is imposed the
current through the sample. From voltage at shunt ter
minal and voltage between voltage electrodes (both
measured in amplitude and phase), and geometric fac
tors, it is obtained
.
ωε+ σ
i
Figure 1 Schematic representation of a soil sample for meas
urement of
in function of frequency.
ωε+σ
i
The field measurements of real soil have inherent dis
persion. A purely mathematical fitting may lead to
physically inconsistent models with quite wrong results,
e.g. by Fourier methods. It is adequate to have a robust
validation criteria of soil models, covering real soil char
acteristics.
In [15] several soil electric models have been pre
sented and justified, which :
 Cover a large number of soil measured parame
ters, with good accuracy, and within the range of
confidence of practical field measurement.
 Satisfy coherence conditions.
In this paper the electrical soil parameters are applied
(σ , ω ε ), in function of frequency, considering a par
ticular set of the models described in [15]. The parame
ters of such models were chosen according to a mini
mum difference criterion for field measured electrical
parameters, in function of frequency, for 68 ground
samples at eight sites, in Brazil, covering very different
soil types and geological structures. The agreement of
obtained models with measured parameters is within or
near the confidence range of field measurement values.
The measurements were carried out in a frequency range
from 100 Hz to 2 MHz . At each site, the maximum dis
tance between ground points at which samples were
collected was less than 500 m.
To show that the influence of small depth surface ef
fects can be neglected in most applications, we indicate,
in Table 1, the soil depth d at which electromagnetic
field related to longitudinal line parameters reduces to
about 5 % of field at soil surface, in four examples, for
three frequencies, f . The first two (examples 1 and 2)
consider, respectively, soil 1 and soil 2 of item II E. of
this paper. The last two (examples 3 and 4) consider
soils similar to examples 1 and 2, but with a low fre
quency conductivity of 1 mS/m.
Table 1 – Soil penetration depth, d , in four examples, for
three frequencies
d [m]
Example 1 Example 2 Example 3 Example 4
21 742 21 218
1 694 1 644
48 164
f = 60 Hz
f = 10 kHz
f = 1 MHz
6 169
510
6 155
477
40 48
B. Soil Models
The models which have been used in the presented re
sults are some of the models described in [15].
With the exception indicated below, the models,
whose results are presented, are a sum of minimum
phase shift parcels, Wj , which apply to the immittance
type magnitude (in complex or tensorial formulation of
alternating magnitudes)
ωεσ
iW
+=
(ω = 2 π f , f being the frequency) ( 1)
where i = + −1 and
∑
=
j
=
m
j
W
0
W
( 2)
All submodels used for Wj are particular conditions of
a Type 3 model described in [1], presented below.
Apart from slow phenomena and hysteresis type phe
nomena, soil behavior is, typically, of minimum phase
shift type. For a great number of soils, on frequency
range ( 0 , 2 MHz ), in a first approach, it is
α
ωσ⋅+=
ba
and
ωε= c
where a, b, c, α are constant parameters (frequency in
dependent).
For some soils, a similar behavior occurs, but for a
smaller frequency range, e.g. ( 0 , 100 kHz ), and for
higher frequencies, the behavior is different, namely
with a lower ωε increase, or until a ωε decrease, when
frequency increases.
In order to analyze the frequency behavior of σ , ε , it
is convenient to consider complex formulations of elec
tromagnetic entities, and to consider σ
mittance. In fact, apart from geometric factors,
may be associated with the admittance of a volume ele
ment δv.
A type 3 model can be described as presented below,
for which :
ωα
( 3)
α
ω⋅
as an im
+σ
ωε+ i
ωε
i
−
⋅=ω
ω
α+α
α
α+α
αα
a i
,,
b i
,,
ab
k)(Wj
, ,
1111
1212
FF
( 4)
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International Conference on Power Systems Transients – IPST 2003 in New Orleans, USA
representing 2F1[…,…,…,…] the hypergeometric func
tion, with four arguments, 2F1 , according to the notation
of [6].
This submodel has four independent parameters (k, α,
a , b).
Considering, in this model (4), a = 0 , the model be
comes :
⋅=
α
ω
+
ω
α
ω
(
α
b
1, 1F12
i ,
, )
b
kWj
=
, 1
+
ω
α
α
b
1F12
i ,
,
1 k
( 5)
Considering, in the model (4) , a = 0 , b → ∞ and
αj = α, the model becomes :
π
ω
2
A parcel Wj as indicated in (6) is equivalent to parcel
α
ω⋅
b
ωε⋅= c
⋅=
tanKj
α
ω
α
j
j
tan +1)(
⋅
⋅=
iKW
jj
( 6)
of σ and to , as indicated in (1) and
j
and αj = α, with the (3), with b = K
α
ω
π
j ,
c
α
2
condition
α
π
2
= tan
b
c
. This condition has been veri
fied in soil measurements, within measurement accuracy
and soil heterogeneity effects.
Considering, in this model (6), αj = 0 , the model be
comes :
σ constant, ω ε null (“pure” conductor) ( 7)
Considering, in the model (6), αj → 1 , the model be
comes :
σ null, ω ε proportional to ω, ε constant (“pure” di
electric) ( 8)
↑ σ σ [mS/m]
Log10 ( f / Hz )Log10 ( f / Hz )
↑ ωεωε [mS/m]
Figure 2 – Electric parameter of soil sample, f in logarithmic
scale.
Within the range ( 0 , 2 MHz ) , for all soil samples
modeled in this paper, it is accurate enough to consider
two parcels, for σ
, one constant (in most cases
real), and the other of type (4) or of type (5), frequency
dependent. In a few cases, there is a net hysteresis effect,
that can be modeled with an imaginary part of the con
stant parcel. For all samples, α is the dominant parame
ter of the relative shape of a frequency dependent parcel,
W
. For α = 0 such a parcel corresponds to
a “pure” conductor ( σ frequency independent, ε null ).
For α = 1 , such a parcel corresponds to a “pure” dielec
ωε+ i
j, of σ
tric ( σ null , ε constant ). In all samples, for a frequency
dependent parcel, it is 0 < α < 1.
ωε+ i
n
α α
Physical soil structure without
major differences among the ten
soil samples at Site 4 .
Figure 3 – Electric parameter of soil sample, f in logarithmic
scale.
C. Soil Samples
In this section it is presented in graphic form σ σ and
ωεωε , in function of frequency, f , for models of one of
the ground samples obtained in Amazon region. In Fig
ure 2 f is represented in logarithmic scale. In Figure 3 it
is represented, in bar form, the distribution of parameter
α of the function relating σ
(5) or (6).
to f , either of Type
ωε+ i
D. Statistical Distribution of Soil Parameters
In order to allow a direct interpretation of statistical
distribution of the main electric parameters of ground, in
a way which is independent of the model details, the
following parameters were chosen, according to the
models adopted, independently, for the 68 soil samples,
satisfying physical coherence conditions:
σ0 = σ (100 Hz) , σ at 100 Hz.
∆r = ∆σ1 = σ (1 MHz)  σ (100 Hz) σ,
increase between 100 Hz and 1 MHz.
∆i = ∆(ω ε)1 = ω ε (1 MHz)  ω ε (100 Hz),
ω ε increase between 100 Hz and 1 MHz.
α parameter of the frequency dependent parcel of
σ + i ω ε.
It was verified that, for these samples, the two parcels
of
ωε+σ
i
, one constant, the other frequency dependent,
are statistically independent. This fact, and the fact that
no significant correlation exists between the pair [∆i ,
α ] , although it exists between the pair [ ∆r , α ] , gives
rise to the hypothesis that:
 The constant and the frequency dependent parcels
of
ωε+σ
i
are related to quite distinct aspects
of physical ground behavior.
 The frequency dependent parcel is mainly associ
ated with a dielectric physical process, with re
lated dissipative effects. Such dissipative effects
are quite different from conductive behavior as
sociated with the constant parcel.
In Figure 4 we represent the probability density, p , of
parameters σ0 , ∆r , ∆i , α , considered separately, and, in
Figure 5, the probability density, p , of parameters [∆i ,
α] , considered together, with Weibull approximations
based on the 68 soil samples.
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International Conference on Power Systems Transients – IPST 2003 in New Orleans, USA
p
σ σ0 [(mS/m) 1]
∆r [(mS/m)1]
p
∆i [(mS/m) 1]
p
p
α α
Figure 4 Probability density, p , of parameters σ0 , ∆r , ∆i , α,
considered separately, with Weibull approximations based in
the 68 soil samples. Scales of p applicable to σ0 , ∆r , ∆i are
graduated in (mS/m)1 .
α α
0.28
0.270.25
0.2
0.15
0.1
0.03
0.01
0.003
0.001
∆i [(mS/m)1]
Figure 5 Probability density, p , of parameters [ ∆i , α ] ,
considered together, with Weibull approximations based in the
68 soil samples and without correlation between ∆i and α .
Values of p , in white, are expressed in (mS/m)1 .
E. Soil Parameters Applied
The soil parameters used in these examples were ob
tained from the experiments described in [3], and are
presented below :
 Soil 1 :
= Α + Β ω
ωε+σ
i
with A, B, α constants, and
A = 84.16 µS/m
B = [0.057849 + 0.12097 i] (µS/m) sα
α = 0.71603
 Soil 2 : σ = Α ; ω ε = 0
which results in ρ : 11882 Ω.m, constant.
The conductivity of the studied soils were chosen to
be equal at low frequency, in order to compare the ob
tained results with the results of traditional procedure
α
that assumes constant conductivity (as measured at low
frequency) and ω ε = 0 . Soil 1 considers two parcels Wj
as described in II.3, namely one constant parcel, A , of
the type (7) and a frequency dependent parcel, Β ωα, of
the type (6). In soil 2, only the first parcel, A, is consid
ered. Although this constant parcel has an apparently
low value, it is not uncommon to measure condutivities
of this magnitude or lower in several Brazilian regions.
By example, for a specific upcoming 1000 kmlong
transmission line, in the center of Brazil: in 20 % of the
line length conductivity was in the range or lower than
the adopted value; the minimum measured value was
near 43 µS/m.
III. CALCULATING THE LINE PARAMETERS
In order to implement the soil model, the line parame
ters were calculated using the approximated formula
which includes the earth effect in longitudinal imped
ance as the equivalent to having an ideal ground surface
at a depth D (complex) below physical ground surface
[7].
'
Dkm
d km
h k
h'k
h m
h'm
k
m
real ground
ideal ground
()
0
1
D
µωεω+σ
=
ii
'
h'k = hk + D'
Figure 6  Conductors k and m position supposing the earth at
a complex depth D’.
The transmission line longitudinal impedance matrix,
per unit length, may be obtained considering:
Z0 =  Z0km  k, m = 1, 2, ..., n
where :
Z0km – longitudinal impedance matrix element, per unit
length;
n – total number of conductors
and
Z0km = Zintkm + Zextkm
where
Zintkm – internal impedance, per unit length, of conduc
tor k, for k = m , 0 for k
Zextkm – external impedance, per unit length, between
( 9)
( 10)
m ;
≠
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International Conference on Power Systems Transients – IPST 2003 in New Orleans, USA
conductors k , m;
and
ωµ
=
km
km
ext
d
D
Zln
2
i
0
π
k, m = 1, 2, ..., n ( 11)
where Dkm and dkm are defined schematically in Figure 6,
and given by :
1
'
µωεωσ +
For the self terms (k = m)
hD'2
=
( 13)
dkm = rk (external radius of conductor k ) ( 14)
and
Zint = Rint + i Xint ( 15)
where
Rint – internal conductor resistance
Xint – internal conductor reactance
In Figure 7 and Figure 8 the per unit longitudinal pa
rameters for the transposed line using both soil models
are presented.
()
i
0
i
=
D
( 12)
kkm
10
1
10
2
10
Frequency [Hz]
3
10
4
10
5
10
6
0,01
0,1
1
10
100
1000
Single threephase transposed line
soil 1  A  84.16 µS/m
B  [0.057849 + 0.12097 i] (µS/m) s
α  0.71603
soil 2  σ = 84.16 µS/m; ε = 0
α
α = β  soil 1
zero  soil 1
α = β  soil 2
zero  soil 2
Resistance [ohm/km]
Figure 7 – Resistance per unit length comparing both soil
models.
10
1
10
2
10
Frequency [Hz]
3
10
4
10
5
10
6
1
10
Single threephase transposed line
soil 1  A  84.16 µS/m
B  [0.057849 + 0.12097 i] (µS/m) s
α  0.71603
soil 2  σ = 84.16 µS/m; ε = 0
α
α = β  soil 1
zero  soil 1
α = β  soil 2
zero  soil 2
Inductance [mH/km]
Figure 8 – Inductance per unit length comparing both soil
models.
The difference between line parameters for the two
soil models is important, namely for the homopolar
mode (e.g., 37 % in the longitudinal resistance per unit
length, at 10 kHz). For fast transients, for which impor
tant frequency range may include frequencies above
10 kHz, the difference between the two soil models may
also be important for nonhomopolar modes. E. g. , for
100 kHz, there is an order of magnitude difference in
resistance per unit length, between the two soil models,
for nonhomopolar modes. Typical cases in which fre
quency range above 10 kHz is important are : transients
originating from lightning; front of wave aspects of tran
sients associated with shortcircuits. Such cases may be
quite important in what concerns insulation coordination.
IV. SINGLE THREEPHASE LINE
APPLICATION
In Figure 9 the data of the threephase line used to il
lustrate the model are presented.
0.4 m
0.4 m
(7.51, 36.00)
(9.27, 24.07)
3.60 m
phase conductors : grosbeak
ground wires : EHS  3/8"
line length : 400 km
sag at midspan
phase cond. : 13.43 m
ground wires : 6.40 m
Figure 9  Schematic representation of the 440 kV threephase
line.
The line parameters were calculated in the range of
10 Hz to 10 kHz. As it is a single line, to represent its
modes (exact ones for a transposed line and quasimodes
for a nontransposed line) Clarke’s transformation ma
trix was applied as explained in [810]. With the longi
tudinal impedance and transversal admittance in mode
domain, the synthetic circuits were calculated, com
posed of one cascade of πcircuits for each mode, each
representing 10 km length. The 10 km length πcircuit
represents properly the line response up to 7 kHz. The
line was supposed ideally transposed.
A. Frequency Analysis
A frequency scan analysis was performed for both
models where the sending terminal had a 1 V source and
the receiving end was open. The relations between the
line ends were analyzed in the range of 10 Hz to 7 kHz.
In Figure 10 the zero sequence response is presented for
the transposed line.
10100 100010000
0
1
2
3
4
5
6
Frequency scan
Transposed line
soil 2 (mod)
soil 1 (mod)
Reception voltage (module) [V]
Frequency [Hz]
Figure 10  Zero sequence  Transposed line.
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International Conference on Power Systems Transients – IPST 2003 in New Orleans, USA
6
The results for both soil models are discussed below :
 The positive sequence response was similar for
both models.
 The zero sequence response for the frequency de
pendant soil model is more damped than the one
which uses only constant conductance.
 The difference between the zero sequence re
sponse for the two soil models is important, e.g.
14 % at 1000 Hz.
V. CONCLUSIONS
In some existing systems there are unexplained differ
ences, sometimes of the order of magnitude, between
calculated and measured values, such as between in
duced voltages, electric field in ground and transferred
voltages. These differences can be related to the soil
models used.
So, it is essential for most applications concerning
grounding systems, or involving electromagnetic phe
nomena affected by ground, to adequately model the
ground behavior, including several aspects not consid
ered in common practice.
For transmission lines, according to specific condi
tions, and the phenomena being studied, it may be quite
important to correctly model the soil, considering fre
quency dependence of
ωε+σ
i
We have presented some illustrative results for a
440 kV threephase transmission line. The soil behavior
is represented through two alternative soil models. In the
first soil model we have considered an accurate soil rep
resentation, satisfying coherence conditions and with
frequency dependent. In the second soil model,
we have considered a constant, frequency independent,
conductance and ω ε much lower than σ. The two mod
els have similar behavior at low frequency, but quite a
distinct one at higher frequency.
In some cases, an adequate earth model can lead to re
sults quite different from those obtained with the usual
procedure of considering the parameter σ of soil fre
quency independent and parameter ε frequency inde
pendent with a relatively small value. The conditions in
which such a difference can be important include the
following examples :
 Switching conditions in which an important homopo
lar component may occur, either due to the spread of
switching of the three poles, or to fault conditions, and
in which the important frequency spectrum is not re
stricted to extremely low frequencies ( < 1 kHz), and
includes frequencies up to about 10 kHz.
 Network sustained operation, faults and maneuvers in
which conditions occur near resonance, for the ho
mopolar component, for frequencies not restricted to
extremely low frequencies ( < 1 kHz), and, e. g. , for
frequency between 1 and 10 kHz.
 Fast transients, for which important frequency range
may include frequencies above 10 kHz. In this case,
the difference between an accurate soil model and
usual assumptions may be important also for non
homopolar modes. Typical cases in which frequency
.
ωε
i
+σ
range much above 10 kHz (for some conditions above
1 MHz) is important are: transients originated by
lightning; front of wave aspects of transients associ
ated with shortcircuits, transients is gasinsulated
substations. From the presented results it is expected
that, for transient phenomena with dominant fre
quency spectrum till above 10 kHz, the distinct ho
mopolar mode response may have quite important ef
fects, not restricted to higher attenuation. By example,
overvoltage shape, namely in front of wave, may be
quite different, with important consequences in insula
tion coordination.
ACKNOWLEDGMENTS
The measurement and modeling of soil samples of which
partial results have been used in this paper, were done under a
Contract of CISCEA, Comissão de Implantação do Sistema de
Controle do Espaço Aéreo (Air Space Control System Estab
lishment Commission) with Fundação COPPETEC. The au
thors thank CISCEA the permission to use such results.
The authors also thank the financial support received from
FAPESP  Fundação de Amparo à Pesquisa do Estado de São
Paulo and from CNPq – Conselho Nacional de Desenvolvi
mento Científico e Tecnológico.
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